I'm just trying to learn induction from scratch and my book asks me to proof the generalized version of the triangle inequality.

If

are arbitrary real numbers, then

**My solution is as follows**:

Let

be the preposition that

and

are true.

is the normal triangle inequality, so i assume i don't need to show that it is true.

With the induction hypothesis i assume that

is true.

Setting

So

We are trying to show that

Using cases

1)

and

this is true because

according to the induction hypothesis. Also according to the transitive property of real numbers if a< b, then a+c < b+c.

2)

and

Since

is true then

3

and

Therefore it is true that

Is my proof correct ?

And is there a better way of doing this?